Computers & Fluids 105 (2014) 244–250

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Computers & Fluids

journal homepage: www.elsevier .com/locate /compfluid

Heat and mass transfer inside micro-tubes with uniform injection

http://dx.doi.org/10.1016/j.compfluid.2014.09.0300045-7930/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: Unité de Recherche Matériaux, Energie et EnergiesRenouvelables, faculté des sciences Gafsa, Université de Gafsa, 2100 Gafsa, Tunisia.Tel.: +216 97325794.

E-mail address: loussif_nizare@voila.fr (N. Loussif).

Nizar Loussif a,b,⇑, Jamel Orfi c

a École Nationale d’Ingénieur de Monastir, Université de Monastir, Monastir, Tunisiab Unité de Recherche Matériaux, Energie et Energies Renouvelables, faculté des sciences Gafsa, Université de Gafsa, 2100 Gafsa, Tunisiac Department of Mechanical Engineering, College of Engineering, King Saud University, Riyadh, Saudi Arabia

a r t i c l e i n f o a b s t r a c t

Article history:Received 25 October 2013Received in revised form 4 September 2014Accepted 15 September 2014

Keywords:Micro-tubeInjectionSlip flowTemperature jump

This work deals with a numerical study of the transport phenomena along a micro-tube subjected to aninjection at the wall. The governing equations expressing the conservation of mass, momentum andenergy with first-order slip velocity and temperature jump boundary conditions were solvednumerically. The numerical model based on the Finite Volume Method was validated using the availabledata. The study reveals significant impact of slip velocity and temperature jump conditions on thehydrodynamic and thermal flows. For very small injection rates, all profiles and quantities remainunchanged compared to the impermeable wall case.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Flows with injection have been widely studied because of theirwide range of applications such as in separation processes and bio-medical technologies. Uniform injection in micro-ducts was stud-ied by Terrill [1], where heat transfer in laminar flow betweenparallel porous plates was investigated. It was found that the fullydeveloped Nusselt number decreases with small injection Rey-nolds number. Pederson and Raithby [2], Kinney [3,4] and othersstudied the heat transfer for laminar flow inside porous tubes. Alarge variation of the Nusselt numbers occurs when consideringinjection. Moussy and Snider [5] investigated the laminar flow overpipes with injection through the porous wall at low Reynoldsnumbers. Analytical expressions describing the two-dimensionalsteady-state laminar flow over an array of porous pipes weredeveloped from the solution of the Navier–Stokes equations forthe case of low wall Reynolds numbers. Libby et al. [6] investigatedthe flow development in a tube with injection of light or heavy gas.Recently, flow field and heat transfer at micro-scale have attractedan extensive research interest due to the rapid development ofMicro Electro Mechanical Systems (MEMS), biomedicalapplications, innovative cooling techniques for integrated circuitsand separation systems.

Modeling the fluid flow with heat and mass transfer for micro-devices is different from that of the macro-scale systems. The ratioof the mean free path to characteristic length known as Knudsen’snumber, Kn = k/L, defines the region where the continuum assump-tion is valid and where it becomes no longer valid for the case ofgases. For small values of Kn, the fluid behavior can be analyzedusing the Navier Stokes equations with no-slip flow boundaryconditions. For values of Kn varying between 0.001 and 0.1 theregime is called slip flow regime [7]. However, for Kn higher than0.1, the continuum description is expected to fail [8]. Othermethods using molecular simulations such as the direct simulationMonte Carlo are used for such ranges of Kn numbers.

Many studies investigated the effect of rarefaction and temper-ature jump condition on the hydrodynamic and thermal fieldsinside ducts. The majority of these studies have been interestedto solve the energy equation when the hydrodynamic flow is con-sidered fully developed for both cases constant wall temperatureand constant wall heat flux [7,9]. It is interesting to mention theanalytical work of Barron et al. [10] where they extended theoriginal Graetz problem of thermally developing heat transfer inlaminar flow through a circular tube to slip flow. Relationships inKnudsen numbers ranging from 0 to 0.12 were developed todescribe the effect of slip flow on heat transfer coefficient.

Few studies have had an interest in studying the impact of slipvelocity and temperature jump on the hydrodynamic and thermalflows development for permeable walls. In fact, Soundalgekar andDivekar [11] studied laminar slip flow through a uniform circularpipe with small suction, and significant impact of slip velocity isshown on the hydrodynamic flow. Singh and Laurence [12] studied

Nomenclature

Cp specific heat at constant pressure (kJ kg�1 K�1)D tube diameter (m)fapp apparent friction coefficient (–)h convective heat transfer coefficient, (W m�2 K�1)k thermal conductivity (W m�1 K�1)Kn Knudsen number (=k/D) (–)L micro-tube length (m)Nux local Nusselt number (=hxD/k) (–)P pressure (Pa)Pr Prandtl number, (=lcp/k) (–)r radial coordinate (m)Re Reynolds number, (=UinqR/l) (–)Rew Reynolds wall, (=VwqD/l) (–)T temperature, (K)U velocity component in the z-direction (m s�1)V velocity component in the r-direction, (m s�1)bV (=V/Vw) (–)

z axial coordinate (m)z⁄ dimensionless axial coordinate, (=z/2DRePr) (–)z+ dimensionless axial coordinate, (=z/2DRe) (–)

Greek symbolsk gas mean free path (m)l dynamic viscosity (Pa s)c ratio of specific heats (–)q fluid density (kg m�3)

Subscriptsin inletw wall1 fully developed

N. Loussif, J. Orfi / Computers & Fluids 105 (2014) 244–250 245

the impact of slip velocity at a membrane surface for ultrafiltrationapplication. Rarefaction has significant impact on concentrationpolarization.

2. Formulation

A simultaneously developing laminar flow inside a micro-tubewith uniform injection is investigated under slip velocity and tem-perature jump boundary conditions. A number of assumptions areapplied: steady state, axi-symmetric and constant fluid propertiesflow with negligible gravitational forces and negligible viscousdissipation.

The geometry and coordinate system for the duct are shown inFig. 1.

Considering the following dimensionless quantities:

�r ¼ rR; �z ¼ z

R; V ¼ V

Uin; U ¼ U

Uin; P ¼ P

qU2in

; T ¼ T � Tw

Tin � Tw

where R is the radius of the tube, Uin and Tin are respectively theinlet velocity and the inlet temperature of the fluid, Tw is the walltemperature.

For a flow inside an isothermal micro-tube, the governingequations and the corresponding boundary conditions innon-dimensional form are:

Fig. 1. Geometry and coordinate system of flow domain.

1r@

@�rð�rVÞ þ @

@�zðUÞ ¼ 0

V@V@�rþ U

@V@�z

!¼ � @P

@�rþ 1

Re@

@�r1�r@

@�rð�rVÞ

� �þ @

2V

@z2

" #

V@U@�rþ U

@U@�z

!¼ � @P

@�zþ 1

Re1�r@

@�r�r@U@�r

!þ @

2U

@z2

" #

V@T@�rþ U

@T@�z

!¼ 1

RePr1�r@

@�r�r@T@�r

!þ @

2T

@z2

" #

At the inlet (z = 0)

U ¼ 1; V ¼ 0; T ¼ 1

Symmetry condition at r = 0

V ¼ 0;@V@�r¼ 0;

@T@�r¼ 0

At r = R

V ¼ Vw ¼ Vw=Uin

U ¼ �2bvKn@U@�r

������r¼1

T ¼ �2btKn@T@�r

������r¼1

At the outlet (z = L)

@U@�z¼ 0;

@V@�z¼ 0;

@T@�z¼ 0

where:

bv ¼ ð2� f vÞ=f v ; bt ¼ð2� f tÞ

f t

2c1þ c

1Pr

The above coefficients fv and ft are known as the tangential momen-tum accommodation coefficient and thermal accommodation coef-ficient respectively. These parameters describing the gas–surfaceinteraction are not only functions of the composition, temperatureand pressure of the gas but also the gas velocity over the surface,the solid surface temperature and roughness. Their values rangefrom around 0 to 1. For most engineering applications, these coeffi-cients are taken to be close to unity [13,14]. The ratio of bt to bv canbe introduced as b. Its value can range from 0 to more than 100 [13].

246 N. Loussif, J. Orfi / Computers & Fluids 105 (2014) 244–250

In this study, bv will be considered as equal to unity. A typical valueof b for many engineering applications is 1.667. This value corre-sponds to fv = 1, ft = 1, c = 1.4 and Pr = 0.7

3. Numerical method and validation

The governing equations with the appropriate boundary condi-tions are solved numerically using the Control Volume Methodand the Simpler algorithm [15]. Based on a grid dependence anal-ysis, the number of nodes used for the simulations was chosen tobe 1000 � 80 in the axial and radial directions respectively. Theproposed model was validated using the available data from liter-ature and analytical solutions when considering an impermeablewall (Rew = 0). As presented in Fig. 2, the axial evolutions of thedeveloping Nusselt number and friction factor corresponding to

Fig. 2. Validation of the numerical model with the results of Jenson [

the continuum model (Kn = 0) are computed and compared withresults provided by Jensen [16] and Hornbeck [17]. In the otherside, fully developed Nusselt number for different degrees of rar-efaction and temperature jump is validated with data provided byCetin et al. [18]. Moreover, the analytical results for the fullydeveloped maximum velocity and friction coefficient as a functionof Kn are presented and compared. Fig. 2 shows an excellentagreement between our results and the previous ones. In all ofthe performed validation tests, the agreements were found tobe fairly good.

4. Results and discussion

The following simulations are performed for fixed Reynolds andPrandlt numbers respectively equal to: 100 and 0.7.

16], Hornbeck [17] and Cetin et al. [18] and analytical solutions.

Fig. 3. Axial velocity evolutions at three positions and for different degrees of rarefaction and different injection Reynolds numbers.

N. Loussif, J. Orfi / Computers & Fluids 105 (2014) 244–250 247

The axial velocity profiles for different values of z/R along theaxis of the micro-tube are presented in Fig. 3. It is important tomention that for small Rew, the axial velocity profiles areunchanged in comparison with the impermeable tube case(Rew = 0) even considering high rarefaction degrees.

For high values of Rew such as Rew = �0.5, increasing injectioninduces an increase of the maximum velocity. Such behavior isdue to mass conservation.

In the other side and when the slip flow condition is applied(Kn is nonzero), the fluid particles adjacent to the solid surfaceof the tube wall no longer attain the velocity of the solid surface.In fact, increasing Kn leads to an increase of the fluid velocity atthe surface which becomes more pronounced when injection is

considered. In the core region of the tube, the fluid acceleratesand its maximum velocity occurring at the centerline of the tubeincreases significantly as a direct consequence of injection effects,while rarefaction tends to decelerate the centerline velocity.

The evolution of the radial velocity along the radius of themicro-tube is plotted in Fig. 4. These profiles being independentof the axial position are affected by rarefaction. For the continuumcase, the maximum radial velocity exceeds the imposed injectionvelocity (i.e. V > Vw) while increasing the Knudsen number, themaximum velocity decreases and reaches the unity at the wallsurface.

Fig. 5 presents the evolution of the centerline velocity for differ-ent degrees of injection and parametrized by Knudsen number. For

Fig. 4. Radial velocity evolution with injection and rarefaction effects.

Fig. 5. Centerline velocity evolution with injection and rarefaction effects.

Fig. 6. Wall velocity evolution with injection and rarefaction effects.

Fig. 7. Axial variation of the apparent friction coefficient for different rarefactionvalues.

248 N. Loussif, J. Orfi / Computers & Fluids 105 (2014) 244–250

impermeable case with continuum flow, the maximum velocityreaches the common value of Poiseuille flow. Increasing injectioninduces an increase of the maximum velocity, while increasing rar-efaction tends to decrease it.

In Fig. 6, the wall velocity is presented. The continuum case ischaracterized by a wall velocity equal to zero even when high val-ues of injection are used. Increasing Kn or/and injection rates leadsto an increase of wall velocity.

Fig. 7 depicts the axial distribution of the friction coefficientwhere injection and rarefaction are taken into account.

For the no-slip case (Kn = 0), the apparent friction coefficient ishigher than that of the slip flow case. It reaches the usual value of16 corresponding to the Poiseuille flow when the flow becomesfully developed.

When a slip-velocity boundary condition is taken into consider-ation, a decrease of the apparent friction coefficient occurs alongthe tube even if injection across the wall is considered. Thisdecrease is very significant at the tube entrance and leads toalmost a uniform distribution of this coefficient along themicro-tube length.

One can also observe that the effect of Rew is weak particularlyin the entrance region of the tube.

Fig. 8. Temperature evolutions at three axial positions and for different degrees of rarefaction, temperature jump and several injection Reynolds numbers.

N. Loussif, J. Orfi / Computers & Fluids 105 (2014) 244–250 249

Small injection values have no significant impact on the fappRecoefficient in comparison with impermeable tube ones.

Fig. 8 mentions the evolution of the temperature profile at dif-ferent locations of the micro-tube. High values of injection havegreat impact on profiles which are accentuated by introducing slipvelocity and temperature jump conditions. Fig. 9 presents the localNusselt’s number distribution along the micro-tube for the case ofslip flow and uniform injection. Here again, the effect of Rew is veryweak even with high values of injection. When considering onlyslip condition, Nu increases along the duct, but the adoption oftemperature jump reduces this heat transfer coefficient.

Fig. 10 illustrates the evolution of the fully developed Nusseltnumber as a function of Rew, Kn, b. Two Prandtl number valuesare considered (Pr = 0.7 and Pr = 1).

The results of Kinney [4] corresponding to the continuum caseare shown and are in a good agreement with the present results.For the case of a continuum model (Kn = 0) and Rew = 0 (imperme-able wall), the fully developed Nusslet number corresponds to theconstant and known value of 3.66. When increasing Pr, the heattransfer coefficient decreases with injection.

Besides and for a fixed Pr, increasing mass injection induces adecrease in the fully developed Nusselt number. In addition, forvery small injection rates, no significant variation on the heattransfer coefficient is observed.

It’s of interest to notice that for a fixed Rew, introducing slipvelocity increases the fully developed Nu, while temperature jumptend to decrease it for values under the continuum case ones.

Fig. 9. Local Nusselt evolution for slip and temperature jump conditions.

Fig. 10. Fully developed Nusselt number as a function of Rew, Kn, b and Pr.

250 N. Loussif, J. Orfi / Computers & Fluids 105 (2014) 244–250

5. Conclusion

A numerical study for a steady-state, two-dimensional, laminarconvective heat transfer in a micro-tube is conducted using first-order slip velocity and temperature jump conditions. The wall ofthe micro-tube is considered permeable allowing the analysis ofthe injection effects. Results for a simultaneously developing flowin a micro-tube for the constant wall temperature (CWT) boundarycondition are presented and analyzed. The main findings of thisstudy can be summarized in the followings:

1. The effect of injection is very weak in the entrance region of themicro-tube and it becomes significant at the tube exit (z = 100R)while very small values of injection (Rew close to zero) keep allquantities unchanged in comparison with the ones relative toimpermeable tube.

2. The hydrodynamic flow is found to be dependant on the degreeof rarefaction and the injection rates. Increasing mass injectioninduces an increase of the maximum velocity and the apparentfriction coefficient, while rarefaction tends to decrease them.

3. The behavior of heat transfer mechanism depends on whetherinjection and/or slip velocity and temperature jump boundaryconditions are applied. It was observed that increasing massinjection induces a decrease of the fully developed Nusseltnumber even considering high rarefaction degrees.

4. For a fixed injection rate, slip velocity increases the fully devel-oped Nusselt number while temperature jump tends todecrease it for values under the continuum case ones.

Acknowledgment

The second author (Dr. J. Orfi) extends his appreciation to theDeanship of scientific research at King Saud University (Researchgroup Project No: RGP-VPP-091).

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